Malmquistindicator

Malmquistindicator, commonly referred to as the Malmquist productivity index (MPI), is a metric for measuring productivity change of a decision-making unit (DMU) between two time periods. It arises from distance-function approaches to data envelopment analysis (DEA) and assesses how performance relative to a production frontier evolves when technology may shift over time. The indicator is named after Sten Malmquist, with later refinements by researchers such as Färe, Grosskopf, Norris, and Zhang.

Calculation of the MPI uses distance functions relative to production frontiers estimated for two time periods. For inputs x and outputs y in periods t and t+1, denote the period-t and period-(t+1) distance functions as D_t and D_{t+1}. The Malmquist indicator for the interval t to t+1 is the product MPI = [D_t(x_{t+1}, y_{t+1}) / D_t(x_t, y_t)] × [D_{t+1}(x_{t+1}, y_{t+1}) / D_{t+1}(x_t, y_t)]. The first term captures frontier movement (technical change), while the second term captures relative efficiency or catch-up to the frontier.

The MPI can be decomposed into two components: efficiency change (catch-up to the frontier) and technical change (shift of the frontier itself). An MPI value greater than one indicates productivity growth between the two periods, driven by improvements in technology, efficiency, or both; a value below one signals deterioration.

Applications and notes: the MPI is widely used to analyze firms, industries, and countries over time and can be computed with input or output orientation and under constant or variable returns to scale. It depends on the choice of inputs and outputs and on data quality; negative values or missing data can complicate interpretation.